$4.$ A small Alberta refinery blends three types of refined petroleum to produce two commercial gasoline products: automobile gasoline and aviation gasoline $($also known as AvGas$).$ The petroleum products used for blending the two gasoline have the following characteristics.

Octane rating is a measure of the quality of a gasoline $-$ the higher the octane rating, the better the performance of the engine. Clearly, aviation fuel will require a higher octane rating than automobile fuel. Sulfur is one of the leading contributors to air pollution. Recent guidelines require sulfur content of fuels to be less than $500$ parts per million.

The following table shows the required characteristics of the two blended gasolines.

For the next three months the refinery has $\backslash (30,000\backslash$mathrm$\{$bbl$\}\backslash )$ of petroleum A available, $100,000$ bbl of petroleum B$,$ and $\backslash (70,000\backslash$mathrm$\{$bbl$\}\backslash )$ of petroleum C$.($A barrel of oil denoted bbl$,$ is equivalent to $159$ litres.$)$ The refinery can sell as much automobile fuel as it blends, but the maximum demand for aviation fuel is $\backslash (20,000\backslash$mathrm$\{$bbl$\}\backslash ),$ since AvGas is only sold to owners of private jets at a small municipal airport.

The estimated average selling price of automobile gas at the pump is expected to be $\backslash (\backslash$$ $0\mathrm{.}85\backslash )$ per litre, and aviation fuel can be sold at $\backslash (\backslash$$ $1\mathrm{.}40\backslash )$ per litre.

$($a$)$ Formulate a linear program to maximize revenue from the sales of auto and aviation gasoline over the next three months. The table below suggests the definitions of decision variables. For example, A$,$ B$,$ and C respectively are the number of bbl of petroleum A$,$ B$,$ and C $.$ V and T are the number of bbl aviation and auto fuel respectively. The combination of two letters, such as AV$,$ represents the amount of petroleum $($A$)$ used to blend the corresponding fuel $($aviation fuel$).$

$($b$)$ Solve the model using Excel.

$($c$)$ Specify the octane rating and the sulfur content of aviation gasoline and automobile gasoline.